91Ó°ÊÓ

The Gaussian integers, \(\mathbb{Z}[i]\), are a UFD. Factor each of the following elements in \(\mathbb{Z}[i]\) into a product of irreducibles. (a) 5 (b) \(1+3 i\) (c) \(6+8 i\) (d) 2

Prove or disprove: Any subring of a field \(F\) containing 1 is an integral domain.

Prove or disprove: If \(D\) is an integral domain, then every prime element in \(D\) is also irreducible in \(D .\)

Let \(F\) be a field of characteristic zero. Prove that \(F\) contains a subfield isomorphic to \(\mathbb{Q}\).

Prove that the field of fractions of the Gaussian integers, \(\mathbb{Z}[i],\) is $$\mathbb{Q}(i)=\\{p+q i: p, q \in \mathbb{Q}\\}$$

A field \(F\) is called a prime field if it has no proper subfields. If \(E\) is a subfield of \(F\) and \(E\) is a prime field, then \(E\) is a prime subfield of \(F\). (a) Prove that every field contains a unique prime subfield. (b) If \(F\) is a field of characteristic \(0,\) prove that the prime subfield of \(F\) is isomorphic to the field of rational numbers, \(\mathbb{Q}\). (c) If \(F\) is a field of characteristic \(p,\) prove that the prime subfield of \(F\) is isomorphic to \(\mathbb{Z}_{p}\)

Let \(\mathbb{Z}[\sqrt{2}]=\\{a+b \sqrt{2}: a, b \in \mathbb{Z}\\}\). (a) Prove that \(\mathbb{Z}[\sqrt{2}]\) is an integral domain. (b) Find all of the units in \(\mathbb{Z}[\sqrt{2}]\). (c) Determine the field of fractions of \(\mathbb{Z}[\sqrt{2}]\). (d) Prove that \(\mathbb{Z}[\sqrt{2} i]\) is a Euclidean domain under the Euclidean valuation \(\nu(a+b \sqrt{2} i)=\) \(a^{2}+2 b^{2}\)

Let \(D\) be an integral domain. Define a relation on \(D\) by \(a \sim b\) if \(a\) and \(b\) are associates in \(D\). Prove that \(\sim\) is an equivalence relation on \(D\).

Let \(D\) be a Euclidean domain with Euclidean valuation \(\nu\). If \(u\) is a unit in \(D\), show that \(\nu(u)=\nu(1)\).

Let \(D\) be a Euclidean domain with Euclidean valuation \(\nu\). If \(a\) and \(b\) are associates in \(D,\) prove that \(\nu(a)=\nu(b)\).